MAT 1502 University of Toronto Fall 2017
MAT 1502 Topics in Geometric Analysis: Convexity
Almut Burchard, Instructor
How to reach me: Almut Burchard,
BA 6234, 8-3318.
- almut @ math.toronto.edu ,
- Lectures W 3:10-4, R 3:10-5pm in BA 6180
(Lectures missed when I travel will be made up on Wednesdays 4:10-5pm).
- Office hours: Monday late-afternoon
About the course: We will explore a variety of topics
in convex and geometry and analysis, moving from classical
theorems and techniques to recent results and open problems.
Possible topics include: Brunn-Minkowski and
isoperimetric inequalities; symmetrization methods; concentration
phenomena; geometric versions of Hahn-Banach;
Legendre transform; extreme points and the
Krein-Milman theorem; monotone and convex matrix functions.
"Asymptotic Geometric Analysis, Part I",
by S. Artstein-Avidan, A. Giannopoulos, and V. Milman.
Mathematical Surveyys and Monographs, Vol. 202 (2015)
- "Convexity: An Analytic Viewpoint", by Barry Simon.
Cambridge Tracts in Mathematics, Vol. 187 (2011)
- "Convex Analysis", by R. Tyrell Rockafellar.
Princeton University Press (1970)
- "Functional Analysis, Sobolev Spaces, and PDE", by H. Brézis
Springer Universitext (2011).
Format: Students enrolled in the course should
work out assignments, and give a one-hour presentation
on a topic of their choice (typically, Wednesdays).
Guests are welcome to participate as they wish.
Week 1 (September 11-15)
- W: Overview
- R: Convex bodies. The Brunn-Minkowski inequality
Week 2 (September 18-22)
- W: Steiner symmetrization
Proof of Brunn-Minkowski (by "competing symmetries")
Week 3 (September 25-29)
The space of convex bodies:
Compactness, Hausdorff vs. Nikodym distance
- R: Two-point symmetrization
and isoperimetric inequality on spheres
Week 4 (October 2-6)
Isoperimetry and concentration on Gauss space
- R: Support function and polar bodies
Week 5 (October 9-13)
- W: Applications of Brunn-Minkowski
- R: Christian Despres:
Classical positions of convex bodies
The University of Toronto is committed to accessibility. If you
require accommodations for a disability, or have any accessibility
concerns about the course, the classroom or
course materials, please contact Accessibility Services as
soon as possible: firstname.lastname@example.org, or